Integers and squares?

Amy was having a conversation with her friend Trina, who had a discovery to share:

Pick any two integers. Look at the sum of their squares, the difference of their squares,
and twice the product of the two integers you chose. Those three numbers are the
sides of a right triangle.

a. Write an equation that models this conjecture using the variables x and y.

b. Investigate this conjecture for at least three pairs of integers. Does her trick
appear to work in all cases, or only in some cases? Explain.

c. Use Trina’s trick to find an example of a right triangle in which all of the sides
have integer length, all three sides are longer than 100 units, and the three side
lengths do not have common factors.

BONUS: If Trina’s conjecture is true, use the equation found in part a to prove the
conjecture. If it is not true, modify it so it is a true statement, and prove the new
statement.